- Research Article
- Open Access
The Existence and Behavior of Solutions for Nonlocal Boundary Problems
© Y. Wang and S. Zheng. 2009
- Received: 16 October 2008
- Accepted: 23 March 2009
- Published: 31 March 2009
The purpose of this work is to investigate the uniqueness and existence of local solutions for the boundary value problem of a quasilinear parabolic equation. The result is obtained via the abstract theory of maximal regularity. Applications are given to some model problems in nonstationary radiative heat transfer and reaction-diffusion equation with nonlocal boundary flux conditions.
- Model Problem
- Black Body
- Nonlocal Boundary
- Chemical Diffusion
- Maximal Regularity
The existence of solutions for quasilinear parabolic equation with boundary conditions and initial conditions can be discussed by maximal regularity, and more and more works on this field show that the maximal regularity method is efficient. Here we will use some of recently results developed by H. Amann to investigate a specific boundary value problems and then apply the existence theorem to two nonlocal problems.
This paper consists of three parts. In the next section we present and prove the existence and unique theorem of an abstract boundary problem. Then we give some applications of the results in Sections 3 and 4 to two reaction-diffusion model problems that arise from nonstationary radiative heat transfer in a system of moving absolutely black bodies and a reaction-diffusion equation with nonlocal boundary flux conditions.
We consider the following quasilinear parabolic initial boundary value problem (IBVP for short):
where is a bounded strictly Lipschitz domain with its boundary and , ,
and is a second-order strongly elliptic differential operator with the boundary operator given by
The coefficient matrix satisfies regularity conditions on , respectively. The directional derivative , is the outer unit-normal vector on ; the function is defined as for ; denotes the trace operator.
We introduce precise assumptions:
where ) are Carathéodory functions; that is, (resp., ) is measurable in (resp., in for each and continuous in for a.e. (resp., . More general, the function can be a nonlocal function, for example, or .
Let and be Banach spaces, we introduce some notations as follows:
(i) , . , .
(ii) for , .
(iii) all continuous linear operators from into , and .
(iv) denotes the Nemytskii operator induced by .
(v) denotes the set of all locally Lipschitz-continuous functions from into .
(vi) , , and , denotes the set of all Carathéodory functions on such that , and there exists a nondecreasing function with
Particularly, is independent of if .
(viii) , ( is the set of integral numbers), is defined as
where , is the dual space of , and is the formally adjoint operator.
(x) if and is an interval in .
(xi) denotes all maps possessing the property of maximal regularity on with respect to , that is, given , the initial problem
has a unique solution .
Now we turn to discuss the local existence result. We write
Exactly, as , where denotes the Banach space of all functions being bounded and uniformly continuous in . So, we will not emphasize it in the following.
where and denote the obvious duality pairings on and , respectively.
After these preparations we introduce the following hypotheses:
(H1) and .
, with , and .
(H3) for some .
Let assumptions (H1)-(H3) be satisfied. Then for each the quasilinear problem (2.1) possesses a unique weak solution for some .
shows the maximal regularity of the operator . By [1, Theorem 2.1], if, for , for some , then the existence and the uniqueness of a local solution will be proved.
On the other hand, the hypotheses guarantee that
This ends the proof.
We see a nonlinear initial-boundary value problem, which, in particular, describes a nonstationary radiative heat transfer in a system of absolutely black bodies (e.g., refer to ). A problem is
3.1. Local Solvability
We assume that (Hr)
(Hr2) is locally Lipschitz continuous and .
Let assumptions (H1)-(H2) and (Hr) be satisfied. Then problem (3.1), for all , has a unique for some .
Hence Theorem 2.1 implies the result immediately.
In fact, Amosov proved in 2005 the uniqueness of the solution for a simple case, that is, problem in which the matrix is independent of (see [2, Theorem 1.4]). In this paper, we also can get the positivity of the solution and the estimates of the solution in and in this part. We have tried to achieve the global existence, but it is still an open problem.
In the rest of this section, we always assume that (H1)-(H2) and (Hr) hold.
(H+) is nondecreasing with , and
Let assumption ( ) be satisfied. If is nonnegative, then the solution of problem (3.1) is also nonnegative.
If , then . The assertion follows.
We denote by the maximal interval of the solution of problem (3.1).
If and , then the solution of problem (3.1) is bounded with its -norm for all .
The claim follows.
One immediate consequence of the above theorem is.
The -norm of the solution , that is, , of problem (3.1) is nonincreasing if .
We now consider the problem (2.1) with the following boundary value condition:
The function in (4.1) can be in nonlocal form.
IBVP (2.1) with a nonlocal term stands, for example, for a model problem arising from quasistatic thermoelasticity. Results on linear problems can be found in [3–5]. As far as we know, this kind of nonlocal boundary condition appeared first in 1952 in a paper  by W. Feller who discussed the existence of semigroups. There are other problems leading to this boundary condition, for example, control theory (see [7–12] etc.). Some other fields such as environmental science  and chemical diffusion  also give rise to such kinds of problems. We do not give further comments here.
Carl and Heikkilä  proved the existence of local solutions of the semilinear problem by using upper and lower solutions and pseudomonotone operators. But their results based on the monotonicity hypotheses of , , and with respect to .
In this section, we assume that (H1) and (H2) always hold and assume that
(Hn1) and for some ;
(Hn2) , satisfies the Carathéodory condition on and .
By the embedding theorem and Theorem 2.1, we get immediately.
Suppose hypotheses of (Hn) satisfy. Then problem (2.1), for all , with defined in (4.1) has a unique for some .
For the simplicity in expression, we turn to consider a problem with nonlocal boundary value
Let assumption (Hk) be satisfied. Then Problem (4.2), for any , has a unique solution for some .
The claim follows immediately from Theorem 4.1.
A special case of problem (4.2) is
That is, and in (4.9) are independent of gradient .
In order to discuss the global existence of solution, in the rest of this section we assume the following.
(Hkl)Suppose there exists a continuous function such that
where for . Therefore, by multiplying with and integrating over , the inequality (4.14) follows the claim.
The boundedness of solution for is used in above deduction.
This closes the end of proof.
4.3. Decay Behavior
In order to investigate the decay behavior of solution for problem (4.9), we assume that
(Hkd) there are two continuous function and ( ) such that
for all .
Let the assumption (Hkd) be satisfied and, be the solution of problem (4.9) with . Then decay to zero as for some small functions .
This ends the proof.
Moreover, one can verify that also decay to zero (as ) if
The first author wishes to thank Professor Herbert Amann for many useful discussions concerning the problem of this paper. The author also want to thank the referees' suggestions. This work is supported partly by the National NSF of China (Grant nos. 10572080 and 10671118) and by Shanghai Leading Academic Discipline Project (no. J50101).
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